Exactly four unweighted death-Birth amplifiers of weak selection on eleven vertices: an exhaustive census of all 1,006,700,565 connected graphs

For death-Birth (dB) updating on a connected, simple, unweighted graph with N vertices, Allen et al. showed that the fixation probability of a single mutant of fitness 1 + δ is 1/N + δ(Neff − 2)/(2N) + O(δ2), and called the graph an amplifier of weak selection if Neff > N. We enumerated all 1,006,700,565 connected graphs on 11 vertices with nauty geng (the count equals OEIS A001349(11)) and decided each with an exact rule: exactly 4 graphs have Neff > 11 (exact rational values ≈ 11.095159, 11.012785, 11.005561, 11.000756), 539 tie at Neff = 11, all regular, and 539 equals OEIS A005177(11), and the remaining 1,006,700,022 are below 11, each certified. The four graphs are not new: by printed Neff, mean degree, minimum degree and normalized-Laplacian λ2 (drawings not compared) they are the four N = 11 graphs of Richter, arXiv:2204.12861v1, p. 8, Fig. 1. What is new, to our knowledge, is that they are all of them at N = 11. The preprint continues our census for N ≤ 10 (no amplifier) and corrects its presentation of Richter's graph as the N = 11 example (see ERRATUM_C1.md). Tier split. Tier A (Lean 4.34.1, Mathlib; axioms within propext, Classical.choice, Quot.sound): for three of the four graphs, the theorems A1_amplifier, A2_amplifier, A3_amplifier (every rational solution of Allen's coalescence system has the stated exact Neff > 11), with kernel-decided negative controls and a planted sorry and axiom caught by the axiom audit; the fourth graph is Richter's, certified in the predecessor preprint. Tier L: Allen's criterion and the step from Neff > N to amplification near r = 1 (their eq. (8)); taken as stated. Tier C: the census count (enumeration completeness rests on geng and the OEIS count; a floating-point proposal is decided by an exact integer residual bound that uses the non-negativity of the inverse of I − M; run with a C port validated against the Python code of the predecessor on N = 3..10 and on a 107-graph N = 11 sample), the exact Neff of R11, and exact-rational sign certificates of the fixation-probability difference against K11 at three values of r for each new graph; the census is not kernel-checked. Controls and a bug. Positive: K11 and C11 tie; Richter's graph is found by enumeration among the 4,721 connected graphs with degrees in [3,4] and 21 edges; the C port reproduces the predecessor census for N = 3..10 (zero amplifiers, ties 1, 2, 2, 5, 4, 17, 22, 167). Negative: a corrupted criterion (pi_i = 1/N) changes the verdicts (9 new candidates at N = 10); the strict test rejects K11, C11 and Richter's parent. The first build of the C port did not escape the backslash that graph6 can contain; this was found and fixed before validation and affects output formatting only. Three observations on Richter's Fig. 1, recorded neutrally and not raised with the author: the caption gives maximum degree 4 for all four graphs, whereas the graph with Neff = 11.0952 has maximum degree 3; and the text names the graph with 11.0128 as having the highest Neff, whereas 11.0952 is higher; and p. 10 prints the mean degree of (b) and (d) as 2.0991, against 2.9091 in the caption. What this is NOT. This is mathematics of graphs and of fixation probabilities in a Moran-type model. It is not a statement about real tumours, tissues, patients or any medical application, and nothing here bears on diagnosis or treatment. It claims no new graph. Limitations. Independently audited by a different AI agent (AUDIT.md, ACCEPT_WITH_CORRECTIONS): Lean recompiled (through a clone of the lean_pinned.py compile path, not lean_pinned.py itself) and kernel-replayed, decision rule re-derived and tested adversarially, N ≤ 10 and 7.27 million N = 11 graphs re-decided with independent code, C port rebuilt reproducibly and rerun on two parts with identical output; no human audit and no statement lock (owed); the C code was reviewed and rerun but not given a line-by-line independent audit; novelty only to our knowledge (a literature search, not a priority claim); the identification with Richter's Fig. 1 is by Neff, mean and minimum degree and λ2, not by drawings; unweighted graphs, dB updating, uniform initialisation and the first-order criterion only; the finite-r checks are three values of r per graph; repository guards were not run. The Elenchus ledger gate exits with code 1 (0 blocking findings, 4 flags LEDGER_UNAUDITED_TIER_A, one per Tier A row).